How the time synchronization was achieved in ancient Greece? The context of this question is about Eratosthenes and his calculation of the size of the Earth based on Sun casting shadows with different length and angle in the same time in two cities ~500 miles apart.
In the particular case of Eratosthenes measuring the radius of the earth, it was done by observing the length of shadows at midday on the summer solstice, in cities that were north-south aligned (to within a few degrees). It was known that the sun was directly overhead on the solstice in the city of Syene in Egypt. But further north in Alexandria on the summer solstice the sun was not directly overhead.
So all he needed to do was identify which day the summer solstice fell on, which was a day that people had been able to determine since before recorded history it seems. Likewise recognising midday was no great difficulty.
You don't even need to do the measurements at the same time - you can do them on successive years if you want, and get the same result.
To add to the answer of PhillS: there was essentially no method of exact time synchronization in two remote places before 17th century. The method of observation at noon at two places only works UNDER THE ASSUMPTION that the places are on the same longitude. The longitudes of Syene and Alexandria are approximately equal, which Eratosthenes knew (when traveling from A to S one faces South, this was probably his justification). Of course this argument can be only approximate.
In fact the time synchronization in different places is EQUIVALENT to the problem of determining longitude. The only general method that was available to the Greeks was observation of Lunar eclipses (which happen at the same time at different places). Except this method, which is also very rough and inconvenient, the general problem of finding longitude (=time synchronization) was not solved until 17 century when Jupiter satellites were discovered. And later, in 18th century two methods were developed: a) very precise observation of the Moon position with respect to the stars, and b) mechanical chronometer.
As PhillS points out, the observations do not have to be made at the same time, but that flexibility is not limited to using observations on successive years, and, as Pieter Geerkens comments, observations at the local noon (either on the same day or the same date in different years of a solar-year calendar) are usable. In either case, however, what you do need is a reasonable measure of the north-south distance between the sites. In principle, this could be achieved by a combination of trigonometric surveying with astronomical measurements to establish the north-south direction. Both these techniques were well known to the ancient Egyptians, and it was the practice to re-survey Egypt annually, to re-establish boundaries after the floods , which gave Eratosthenes the distance, though he may have simply assumed that Syene was approximately due south of Alexandria.
Once you have a value for the circumference of the Earth, and assume it to be approximately spherical, you have sufficient information to determine the local-time offset between any locations linked by a reasonably accurate two-dimensioned survey, but the ancients did not really have any practical need for that information.